Detecting Students' Experiences of Discontinuities Between Middle School and High School Mathematics Programs: Learning During Boundary Crossing

Abstract

Transitions from middle school to high school mathematics programs can be problematic for students due to potential differences between instructional approaches and curriculum materials. Given the minimal research on how students experience such differences, we report on the experiences of two students as they moved out of an integrated, problem-based mathematics program in their middle school into a high school mathematics program that emphasized procedural fluency. We conducted an average of two interviews per year for two and a half years with participants and engaged in participant-observation at their high school. In this study, we illustrate an analytic process for detecting discontinuities between settings from participants' perspectives. We determined that students experienced a discontinuity if they reported meaningful differences between settings (frequent mention, in detail, with emphasis terms) and concurrently reported a change in attitude. Additionally, these students' experiences illustrate two opportunities to learn during boundary-crossing experiences: identification and reflection.

ACKNOWLEDGMENTS

This project was supported by a grant from the National Science Foundation to the third author (REC-9903264). In addition to the authors, the Mathematical Transitions Project team played a role in the work reported here. The other members of this team were, in alphabetical order, Dawn Berk, Carol Burdell, Gary Lewis, Violeta Rosca-Yurita, and Jon Star.

We are grateful to the students in this study for their willingness to share their experiences with us.

Notes

¹By “problem-based,” we mean that students' learning is facilitated by engaging in problems, in contrast to applying previously mastered skills to problems (Hiebert et al., 1996).

²Although curriculum materials have been written through funding from the National Science Foundation [NSF] to support putting the NCTM Standards into practice, there has not been consistent agreement that these materials appropriately support students' learning. Evidence suggests that that teachers who use NSF-funded curriculum materials can effectively develop students' procedural skills and conceptual understanding (e.g., Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; Reys, Reys, Lapan, Holliday, & Wasman, 2003; Riordan & Noyce, 2001). However, mathematicians (e.g., Wu, 1997) have expressed concerns, suggesting that the materials devalue logical arguments and overemphasize real-world applications.

³All names are pseudonyms.

⁴UCSMP has been considered to be aligned with the NCTM Standards (1989, 1991, 2000) due to increased emphasis on real-world uses of mathematics and multistep problem solving (Thompson & Senk, 2003), but the implementation of this curriculum at this high school was aligned more closely with traditional instructional approaches.

⁵Note that researchers' observations were not used as a primary data source. Researchers observed to inform the construction of follow-up questions to pose during interviews and to build rapport with participants while observing their classrooms. Researchers' observations indicated that students' reports during interviews aligned with our observations.